Let $S$ be a non-empty set of positive integers such that for any $n \in S$, all positive divisors of $2^n+1$ are also in $S$. Prove that $S$ contains an integer of the form $(p_1p_2 \ldots p_{2023})^{2023}$, where $p_1, p_2, \ldots, p_{2023}$ are distinct prime numbers, all greater than $2023$.